```Date: May 5, 2013 11:53 AM
Author: ross.finlayson@gmail.com
Subject: Re: Matheology § 258

On May 5, 7:50 am, fom <fomJ...@nyms.net> wrote:> On 5/5/2013 9:27 AM, Julio Di Egidio wrote:>>>>>>>>>> > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote in message> >news:33cbfa02-27a6-459d-9b0b-330765e81d98@zo5g2000pbb.googlegroups.com...>> >> Almost seems like:  making indices N x N, has that for the one image,> >> each element is finite yet unbounded, while for the other it is> >> infinite.  This basically gets into considering a copy or instance of> >> N, then another, has in some manner the sputnik of quantification or> >> here correlation among the two, that for the same specification, one> >> has finite and unbounded elements, the other finite and unbounded and> >> as well: infinite elements.>> > Indeed an infinite set is a set, i.e. an actual (i.e. completed, in the> > math sense) infinity, while N is the potentially infinite.  My hunch is> > that we should be using N* (the compactification of N, to begin with) as> > the counting set outside the finite realms: then arithmetic and set> > theory could indeed be equivalent.>> Isn't that what modern logic forces upon us?>> To interpret the universal quantifier for the> Peano axioms under the received paradigm, the> natural numbers have to be a totality.>> In Markov, the potentially infinite is described> relative to a specific set of definitions for> constructed syntax and a specific recharacterization> of quantifiers with respect to those constructs.> In view of such careful reasoning, simply attributing> potential infinity to |N without consideration of> what is involved with the interpretation of quantifiers> seems as if it is missing something.What is the set of natural integers?  Sets are defined by theirelements, so it is a set that satisfies the predicate contains(n) foreach n that is a natural integer.  Then, in a pure set theory thereare only sets:  what set is a natural integer?  Generally the notionis that the finite ordinals are defined, with a constant zero and arule to generate for each a successor, then that the set of finiteordinals is an inductive set, where the set has the property that ininduction, the ordinal elements alpha can be used to build inductivecases that for each ordinal alpha there exists a unique and distinctordinal alpha+1 that is not the successor of any previous number.But, that is not so clearly all that a natural number is, an ordinal,though the ordinals are given labels matching those of the naturalintegers.  Natural integers have arithmetic defined, to define the PAor PA (Presburger (+) or then Peano (+,*) Arithmetic), while thelabels uniquely identify elements of the domain and range of thoseoperations as functions, addends and sums, multiplicans and products,the establishment of the values of sums is a consequent raft oftheorems that:  go into the definition of natural integers, thus, intoany pure set they are in as to what elements they are.Then the notion as above is that to have elements of the naturals andinductive set identify distinct elements, and then to have a copy ofan inductive set indicate elements of those elements, basicallybuilding the space of rows and columns instead of just induction, hasthen built a sequence and not just course-of-passage, and as thecomplement exists an omega-sequence, with the compactification of N(point at infinity) thus resulting as a theorem instead of as adefinition as it is in ZFC, with the only constant besides 0, theordinals, being omega, the next limit ordinal and itself defined asthe collection of finite ordinals.Basically then is a consideration that the operation of a set that isan ordinal alpha to successor is then to the closure of a set w thatis of finite ordinals to correlation with sets of finite ordinals,that builds systems with the completed infinity from the potential:back into the language of the collection of elements.  Then, well-foundedness aka regularity isn't necessarily a true axiom, not that itis anyways from pure naive set theory, but that N e N isn'tparadoxical because it is only from building up the systems to that,variously:	the naturals are implicitly compact	there are infinite elements in the naturalsThese features of the numbers, in their entire collection, aren't sodirectly relevant to most one-off computations of finite arithmetic ortheir unbounded associations.   Then where that is ignored, having thenatural numbers as sets as ordinals, is a reasonable abstraction tokeep the overall machinery out of the way of simple computation.https://en.wikipedia.org/wiki/Category:Elementary_arithmeticIn pure set theory, sets are defined by their elements and containonly sets.  In number theory, numbers are indviduals generally builtfrom the natural integers, and all the operations on them andrelations among them, and as to equality of the values of the numbersin extended number systems toward the continuum of real numbers, with1 = 1.0, and so on.  Then, as pure sets, numbers are all those things.Elements of an inductive set, ordinals that contain their predecessor,and naturals, aren't interchangeable.   Ordinals and naturals canserve as representations of elements of an inductive set, and invarious cases as representations, or labels, or names, of eachother.Then as above where the sequence is the representation, in the 1-aryrepresentation of a value, succession was used to build the sequences,then to go over all of them and build a different sequence, puts theomega-sequence into the language, of the finite sequences.Then, Cantor's result is interpreted not as making the collection offinite sequences incomplete, instead, introducing the compactificationto then:  that of the language of the representations, the resultingelement of that representation, is a point at infinity.  Basicallythen this is the notion of having a construction of omega, as a setand not necessarily and indeed not a well-founded set, instead ofdefining it as a constant in the language which would then beinconsistent with the resulting construction.Then number theorists might find it easy to work up using sets forcontaining numbers, in provisional set theory where the notion of setsis just for quantifiers over elements satisfying predicates and as toinduction:  that the natural integers can have a point at infinitywithout breaking ZFC.Regards,Ross Finlayson
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